Multi-period structure of electro-weak phase transition in the 3-3-1-1 model

The electroweak phase transition (EWPT) is considered in the framework of 3-3-1-1 model for Dark Matter. The phase structure within three or two periods is approximated for the theory with many vacuum expectation values (VEVs) at TeV and Electroweak scales. In the mentioned model, there are two pictures. The first picture containing two periods of EWPT, has a transition $SU(3) \rightarrow SU(2)$ at 6 TeV scale and another is $SU(2) \rightarrow U(1)$ transition which is the like-standard model EWPT. The second picture is an EWPT structure containing three periods, in which two first periods are similar to those of the first picture and another one is the symmetry breaking process of $U(1)_N$ subgroup. Our study leads to the conclusion that EWPTs are the first order phase transitions when new bosons are triggers and their masses are within range of some TeVs. Especially, in two pictures, the maximum strength of the $SU(2) \rightarrow U(1)$ phase transition is equal to 2.12 so this EWPT is not strong. Moreover, neutral fermions, which are candidates for Dark Matter and obey the Fermi-Dirac distribution, can be a negative trigger for EWPT. However, they do not make lose the first-order EWPT at TeV scale. Furthermore, in order to be the strong first-order EWPT at TeV scale, the symmetry breaking processes must produce more bosons than fermions or the mass of bosons must be much larger than that of fermions.

Keywords: Spontaneous breaking of gauge symmetries, Extensions of electroweak Higgs sector, Particle-theory models (Early Universe)

I. Introduction
The EWPT is another view of spontaneous symmetry breaking in Theoretical Particle Physics. The latter is a transition of the Higgs field with vanishing VEV to non-zero one.
The EWPT plays an important role at early stage of expanding universe; and its issue is also related to hot topics such as Dark Matter (DM) or Dark Energy. From a micro viewpoint and within the current limits, candidate for DM may be a heavy particle. If we accept the symmetry-breaking mechanism as an universal mechanism, then mass of the DM candidate must also be generated through a phase transition process. Moreover, if the mass of the DM candidate is very large so the phase transition process must take place before the EWPT of the Standard Model (SM) and must also follow the gradually decreasing temperature structure of the universe.
As in the SM, the EWPT process has only one phase at the energy level around 200 GeV. This process is accompanied by mass generation of particles. However, at present, the existence of heavy particles is possible only at energy scale larger than 200 GeV. Therefore, the production of these heavy particles interacting with the SM ones, must also be considered.
At present, the mechanism of symmetry-breaking is believed to be accurate, but the Higgs potential is not exactly determined because its form is model dependent.
The EWPT consists of an important question of phase transition which must be a strongly first-order phase one. This is the third Sakharov condition being deviation from thermal equilibrium [1]. The mentioned condition together with B, C, CP violations leads to solution of the Baryon Asymmetry of Universe (BAU). The B, C and CP violations can be seen throughout the sphaleron rate and the CKM-matrix in models [2] or other CP violation sources as neutrino mixing.
At present, the EWPT is considered at a one-loop level, particularly, in beyond the Standard Model. A new trend nowadays is multi-phase calculations in multi-Higgs scalar potential.
In order to consider the EWPT, we must build the high-temperature effective potential which is usually in the following form where v is the VEV of Higgs boson. The first order EWPT binds that the strength of phase transition should be larger than the unit (S = vc Tc ≥ 1, where v c is VEV of Higgs field at a critical temperature T c ).
The effective potential V ef f in Eq. (1) is a function of temperature and VEVs. It can have one or two minimums when the temperature goes down. At T c , the two minimums are separated by a potential barrier, the VEV of Higgs field crosses over from vanishing VEV to a non-zero VEV. This transition is called the first order phase transition and it can cause large deviations from thermal equilibrium.
The EWPT has been calculated in the SM [2] and in some extended models [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. It is reminded that DM, heavy particles and neutrino oscillations can be triggers of the EWPT [18]. The most studies of the EWPT are in the Landau gauge. However gauge also made contributions in EWPT as done in Ref. [17]. It is reminded that in some extended models, Higgs sector consists multi-vacuum structure of which the classical example is the Two Higgs Doublet model. This additional Higgs structure can be a new source to answer the BAU puzzles.
Another example of multi-vacuum structure belongs to the models based on SU(3) C ⊗ SU(3) L ⊗U(1) X group [19,20] called 3-3-1 models for short. There exist two main versions of the 3-3-1 models: the minimal [19] and another with right-handed neutrinos [20]. To provide an explanation for the observed pattern of SM fermion masses and mixings, various 3-3-1 models with flavor symmetries and radiative seesaw mechanisms have been proposed in the literature 1 . However some of them involve non-renormalizable interactions. In addition the 3-3-1 models do not give completely desired answer on the DM issue. In the recently proposed group has a good advantage in explaining DM. Phenomena of this model such as DM, inflation, leptogenesis, neutrino mass, kinetic mixing effect, and B − L asymmetry, have been studied in Refs. [24][25][26][27][28]. The 3-3-1-1 model has three Higgs triplets to generate masses of fermions and the mass of new heavy particle with masses around some TeVs. This model fits with candidates for DM. The presence of the above mentioned particles might also lead to interesting consequences such 1 With the help of discrete Z N symmetries, the 3-3-1 model with β = 1 √ 3 can provide solutions of neutrino mass and mixing, DM and inflation [21,22] as the baryon asymmetry or EWPT which is a subject of this study.
This article is organized as follows. In section II, the matter fields and Higgs bosons in the 3-3-1-1 model are briefly reviewed. In section III, the effective potential having the contribution from heavy bosons a function of temperature and VEVs is derived. In section IV, we analysis in details structure of phase transition, find the first order phase transition and show constraints on mass of charged Higgs boson in the case without neutral fermions.
In section V, we discuss the role of neutral fermions in the EWPT problem. Finally, we summarize and make outlooks in section VI.
II. Brief review of the 3-3-1-1 model Nowadays, heavy particles are widely accepted to be exist. Within their existence, unexplained problems can be caused. They may be a candidate for DM, or just new ones. The 3-3-1-1 model has many new particles inserted in the multiplet of the gauge group where U(1) X is the gauge group associated with the electromagnetic interaction and U(1) N is the gauge group associated with the conservation of B − L number when combining with SU(3) L charges [23][24][25][26][27].
To keep the model being anomaly free, the fermion content has to have equal number of the SU(3) L triplets and anti-triplets as follows [23] where a = 1, 2, 3 and α = 1, 2 are family indices. N aR is neutral fermions playing a role of candidates for DM. In (2), the numbers in bracket associated with multiplet correspond to number of members in the SU(3) C , SU(3) L assignment, its X and N charges, respectively.
The Higgs sector of the model under consideration contains three scalar triplets and one singlet as follows ρ = ρ + 1 , ρ 0 2 , ρ + bottom element in lepton triplet (N aR ) without lepton number, ones have [23] B Note that in this model, not only leptons but also some scalar fields carry lepton number as seen in Table I Particle ν e N U D η 3 ρ 3 χ 1 χ 2 φ From Table I, we see that elements at the bottom of η and ρ triplets carry lepton number −1, while the elements standing in two first rows of χ triplet have the opposite one +1.
To generate masses for fermions, it is enough that only neutral scalars without lepton number develop VEV as follows For the future presentation, let us remind that in the model under consideration, the covariant derivative is defined as where G iµν , A iµν , B µν , C µν and g s , g, g X , g N correspond gauge fields and couplings of SU(3) C , SU(3) L , U(1) X and U(1) N groups, respectively.
The Yukawa couplings are given as From Eq. (8), it follows masses of the top and bottom quarks as follows while masses of the exotic quarks are determined as The Higgs fields are expanded around the VEVs as follows It is mentioned that the values u and v provide masses for all fermions and gauge bosons in the SM, while ω gives masses for the extra heavy quarks and gauge bosons. The value Λ plays the role for the U(1) N breaking at high scale; and in some cases, it is larger than ω.
The scalar potential for Higgs fields is a function of eighteen parameters When constructing this Higgs potential, triple scalar self-interactions needs to be limited because it forces us to introduce a f parameter (f has a mass dimension the same as ω) that can like an interrupt factor for these interactions. In addition, f can be replaced by one Higgs field or another interaction among three Higgs fields. Thus, the mentioned interaction will become a fourth-or sixth-order coupling. We often do not consider high-order interactions (because these high-order interactions may be difficult to renormalization. However they may be related to other hypothetical offending processes). Therefore we can ignore f in this article though it may have a different role in other problems. For detailed analysis of the Higgs sector in the model under consideration, the reader is referred to Ref. [23].
Returning to our work, in order to limit the parameter number, as above mentioned, we will ignore f hereafter.

A. Higgs boson masses
Substituting Eq. (9) into Eq. (10) yields where V 0 and V 1 are the minimum interaction term being independent of scalar fields and linear dependent on fields, respectively: Hence, the potential minimization conditions are obtained by From (11) we get the part for charged Higgs bosons From the above equations, after some manipulations, the mass terms of charged Higgs bosons are given by where Similarly, the part of neutral Higgs bosons is given by: Combination among S χ and S ′ η yields where physical boson H 3 is given by The mass of neutral Higgs bosons is presented in Table II Neutral Higgs boson Squared mass 2λΛ 2 λ 9 ω 2 2 λ 9 u 2 2 2λ 3 u 2 2λ 2 ω 2 2λ 1 v 2 λ 9 (u 2 +ω 2 ) 2 Remind that the massless Goldstones bosons are: X A 4 , A η , A ′ χ , A ρ in neutral scalar sector and two massless combinations orthogonal to the charged Higgs bosons. It is noted that at the limit f −→ 0, the results given in [24][25][26] are consistent with those of this study.

B. Gauge boson sector
The gauge bosons obtain masses when the scalar fields develop the VEVs. Therefore, their mass Lagrangian is given by Substituting the scalar multiplets η, ρ, χ and φ with their covariant derivative, we obtain where we have denoted t X ≡ g X g , t N ≡ g N g , and The mass Lagrangian can be rewritten as [24][25][26][27] L gauge mass = where the Lorentz indices have been omitted and should be understood. The squared-mass matrix of the neutral gauge bosons is found to be: The non-Hermitian gauge bosons W ± , X 0,0 * and Y ± are physical fields with corresponding masses: Because of the constraints u, v ≪ ω, we have m W ≪ m X ≃ m Y . The W boson is identified as the SM W boson. It follows   Table III From aforementioned analysis, it follows that the phenomenological aspects of the 3-3-1-1 model can be divided into two pictures corresponding to different domain values of VEVs.
The physical neutral gauge bosons are derived through the following transformation ( The above diagonalization is realized through three steps [24][25][26][27], The first step: The second step: The final step: In Eq. (25), the E is a two-component vector given by [24][25][26][27] Finally we obtain the masses of neutral gauge bosons as follows From the experimental data ∆ρ < 0.0007, ones get u/ω < 0.0544 or ω > 3.198 TeV [23] (provided that u = 246/ √ 2 GeV as mentioned). Therefore, the value of ω results in the TeV scale as expected.

Picture (ii):
If we assume Λ ≫ ω ≫ u ∼ v, three gauge bosons are derived as [24][25][26][27] From the Table (III) and Eqs. (29,30,31), the W ± boson and the Z boson are recognized as two famous gauge bosons in the SM. Now we turn to the main object -the effective potential.

III. Effective potential
Within the above assumption, the Higgs potential is given as follows [23][24][25][26][27], from which, ones obtain V 0 depending on VEVs : Here V 0 has quartic form like in the SM, but it depends on four variables φ Λ , φ ω , φ u , φ v , and has the mixing terms between them. However, developing the potential (32), we obtain four minimum equations. Therefore, we can transform the mixing between four variables to the form depending only on φ Λ , φ ω , φ u and φ v . Hence, we can write In order to derive effective potential, we need the mass spectrum of fields. Starting from the Lagrangian of the scalars (both kinetic and potential terms) and Yukawa interactions, and expanding Higgs fields around VEVs, we obtain the mass terms for all fields in the 3-3-1-1 model.
The gauge sector in the 3-3-1-1 has ten gauge bosons: the photon and nine massive gauge bosons.
The latter includes two massive like the SM Z and W ± bosons, and two new heavy neutral Z 1 , Z 2 bosons, the charged gauge bosons Y ± and the neutral non-Hermitian bosons: X 0,0 * . The Higgs sector contains four charged Higgs H ± 1 , H ± 2 , seven neutral Higgs  Table IV.
From the mass spectra, we can split masses of particles into four parts as follows Taking into account Eqs. (33) and (34), we can also split the effective potential into four parts (26) Eq. (27) Eq. (28) Picture (ii)

Complex Higgs boson Complex Higgs boson
Squared mass It is difficult to study the electroweak phase transition with four VEVs, so we assume φ Λ ≈ φ ω , φ u ≈ φ v over space-times. Then, the effective potential becomes

IV. Electroweak phase transition without neutral fermion
Taking phase transitions in this model into account, it is important to find the activity domain of ω, Λ, u and v. Looking at data in Ref. [30,31], we arrive to assumption: m Z 2 ≥ 2.2 TeV. In addition, from Ref. [23], we also assume m Z 2 < 2.5 TeV. Hence From the constraint in (35), we will infer the domain values of ω and Λ. It is worth mentioning that in the 3-3-1-1 model, the structure of symmetry breaking which can be divided into two or three periods depending on scale of VEVs as suggesting in the above two pictures.
When our universe has been expanding and cooling due to u scale, the symmetry breaking SU (2) → U (1) is turned on, which generates masses of the SM particles and the last part of masses of H 2 , H 3 , X ± , Y ± . Therefore, phase transition SU (2) → U (1) only depends on φ u ∼ φ v . (2) This phase transition involves exotic quarks, heavy bosons, but excludes the SM particles. As a consequence, the effective potential of the EWPT SU (

Phase transition SU (3) → SU
Applying the Coleman-Weinberg's method, the effective potential V ef f (φ ω ) is given as where and The minimum conditions are The values of V ef f (φ ω ) at the two minima become equal at the critical temperature and the phase transition strength are From Eqs. (26,27,28), with the limit of m Z 2 given in Eq.(35), it follows: 5.856 TeV ≤ ω ∼ Λ ≤ 6.654 TeV.
In this work, we assume ω = 6 TeV, so that m Z 1 = 8.304 TeV and m Z 2 = 2.254 TeV. The problem here is that there are nine variables: the masses of U, D 1 , D 2 , H 2 , H 3 and A ′ η , S ′ χ , S 4 , Z 1 .
However, for simplicity, we assume m Consequently, the critical temperature and the phase transition strength are the function of O and P ; therefore we can rewrite the phase transition strength as follows In Figure  The mass region of particles is the largest at S ω = 1, the mass region of charged particles and neutral particles are From Eq. (44) it follows that the maximum of S ω is around 70.

Phase transition SU (2) → U (1)
In this period, the symmetry breaking scale equals to u = 246/ √ 2 and the masses of the SM particles and apart of masses of X 1 , X 2 , X, Y, H 1 , H 2 , H 3 , A χ , S η are generated. The effective potential of EWPT SU (2) → U (1) is given as The minimum conditions are where The mass region of neutral and charged particles given in Table (V) leads the maximum phase transition strength which must be 2.12. This is larger than 1 but the EWPT is not strong. In picture (ii), m 2 Z 2 ≃ g 2 c 2 W ω 2 (3−4s 2 W ) with the limit of m Z 2 given in Eq. (35), we obtain 5.53 TeV ≤ ω ≤ 6.3 TeV. Therefore, we also assume ω = 6 TeV in this picture.
Because Λ ≫ ω = 6 TeV and ω ≫ u ∼ v, therefore there are three periods. The first process (1) in the picture (i).
The first process is a transition of the symmetry breaking of U (1) N group. It generates mass for Z 1 through Λ or Higgs boson S 4 . The third process is like the SU (2) −→ U (1) EWPT in picture (i). The second process is like the SU (3) −→ SU (2) in picture (i) but it does not involve Z 1 and The second process has the effective potential is like Eq. (36). In addition, parameters and the minimum conditions are like Eqs. (37,39,40,41,42,43)  In our numbering process, when we import real T C , the mass region of charged and neutral particles are The mass region of charged bosons is narrower than that in Fig. 1. From Eq. (44), the maximum of S has been estimated to be around 100.

V. The role of neutral fermions in EWPT
The masses of N R can be generated by the scalar content by itself via an effective operator [23] invariant under the 3-3-1-1 symmetry and W -parity: The mass scale of N R is unknown, however it can be taken in TeV scale. However, when analyzing the scattering of N R with distributions of X, Y, Z 2 bosons as given in [23], the mass of N R is drawn as follows In the SU (3) −→ SU (2), if we add the contribution of neutral fermions, the maximum of S would decrease but the neutral fermions does not make lose the first-order EWPT, i.e., as our estimation in Table (VI) Period (2)   In Table VI, we only estimate the maximum strengths and show that these maximum values are significantly reduced. However, it is very difficult to accurately calculate these values because the problem has many parameters (the mass of heavy particles) and these values can change slightly (but not too much) with different approximates. Table VI, when the absence of neutral fermions, in the picture (i) , Z 1 is involved in the SU (3) −→ SU (2) EWPT; while S = 2E λ Tc , contributions of Z 1 make increasing E and λ, but λ increase stronger than E. In the picture (ii), Z 1 is not involved in the SU (3) −→ SU (2) EWPT. Therefore S max of the picture (ii) (100) is larger than S max of the picture (i) (70).

More in
Furthermore, when neutral fermions are involved in both two pictures, S max in picture (ii) decreases faster than S max in picture (i). Because there is a tension between Z 1 and neutral fermion in the picture (i); the same thing is not have in the picture (ii) in the absence of Z 1 .
If neutral fermions follow the Fermi-Dirac distribution (i.e., they act as a real fermion but without charge), they increase the value of the λ and D parameters. Thus, they reduce the value of strength EWPT S because S = E 2λ Tc and E do not depend on neutral fermions. This suggests that DM candidates are neutral fermions (or fermion in general) reduce the maximum value of the EWPT strength.
However, the EWPT process depends on bosons and fermions. Boson gives a positive contribution (obey the Bose-Einstein distribution) but fermion gives a negative contribution (obey the Fermi-Dirac distribution). In order to have the first order transition, the symmetry breaking process must generate mass for more boson than fermion.
In addition, in this model, the neutral fermion mass is generated from an effective operator.
This operator which demonstrates an interaction between neutral fermions and two Higgs fields.
The above neutral fermion is very different from usual fermions. The M parameter has an energy dimension, and it may be an un-known dark-interaction. Thus, neutral fermions only are effective fermions, according to the Fermi-Dirac distribution, but its statistical nature needs to be further analyzed with other data.

VI. Conclusion and outlooks
It is known that the mass of Goldstone boson is very small [31] and the physical quantities are gauge independent so the critical temperature and the strength is gauge independent [17] or the survey of effective potential in Landau gauge is also sufficient, or other word speaking, do not need in any gauge. Therefore, in the Landau gauge, the structure of EWPT in the 3-3-1-1 model with the effective potential at finite temperature has been draw at the 1-loop level, this potential has two or three phases.
The self-interactions of Higgs, with f parameter, in this model, that we do not consider in both two or three phase transitions. Thus calculating the corrections with f can reveal many new physical phenomena. In addition, from the phase transitions, we can get some bounds on the Higgs self-couplings.
In conclusion, the model has many bosons which will be good triggers for first-order EWPT.
The situation is that as less heavy fermion as the result will be better. However, strength of EWPT can be reduced by many bosons (such as Z, Z 1 , Z 2 in the 3-3-1-1 model).
Although we only work on the 3-3-1-1 model, but this calculation can still apply to other models with multi-period EWPT. Our next works will calculate CP violations and correction of neutral fermion-dark matter, in order to analysis in details baryogenesis.